A Supply Chain Network Economy Modeling and Qualitative Analysis

8/6/2019 A Supply Chain Network Economy Modeling and Qualitative Analysis

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1. Introduction

This paper proposes a modeling intitiative that is network-based to for-malize both intra supply chain cooperation and inter supply chain competi-tion. In particular, we propose a general network model for a supply chaineconomy since it is now recognized that when it comes to production, dis-tribution, and consumption today, it is no longer simply firms that competewith one another, but, rather, it is supply chains, consisting of economicdecision-makers and their spectrum of activities. The supply chain networkeconomic model that we develop in this paper aims to answer such funda-mental questions as: How do economic decision-makers form coalitions in acompetitive environment? How do supply chains compete with one another,

and how does one identify the winning supply chain?We note that, to-date, the major emphasis in the study of supply chains

in the literature has been the optimization of a supply chain that is owned(and controlled) by a single entity with various issues explored such as: thedistribution network design, coordination, and inventory management (see,e.g., Chen (1998) and the reviews by Geoffrion and Power (1995) and Beamon(1998)). More recently, competition among supply chain agents has been ad-dressed using game theoretic formulations as in the inventory-capacity gamemodels of Cachon and Zipkin (1999) and Cachon and Lariviere (1999a, b)who studied two stage serial supply chains. Corbett and Karmarkar (2001),

on the other hand, investigated a supply chain network consisting of severaltiers of decision-makers who compete within a tier. Under the assumptionof identical linear production cost functions, they determined the number ofentrants under Nash equilibrium.

Nagurney, Dong, and Zhang (2002), in turn, developed a three tieredsupply chain network equilibrium model consisting of manufacturers, retail-ers, and consumers, who may compete within a tier but cooperate betweentiers. Using a variational inequality formulation, the equilibrium productionoutputs, shipments, and consumption quantities, along with the prices of theproduct associated with the different tiers could then be determined with aniterative computational procedure. Nagurney et al. (2002a) further extended

the basic framework to incorporate electronic commerce whereas Nagurney etal. (2002b) focused on the dynamics of supply chain networks. Subsequently,Dong, Zhang, and Nagurney (2002) developed a supply chain network equi-librium model with random demands associated with the product at different

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retail outlets and formulated and solved the model as a variational inequalityproblem. Our work is similar in spirit, although derived from an entirely dif-ferent perspective an application setting, to that of Dafermos and Nagurney(1984), who constructed a general network model of production, along withits variational inequality formulation.

This paper is organized as follows. In Section 2, we introduce the conceptof a supply chain economy and develop the general network model. In par-ticular, we consider an economy with multiple profit-driven decision-makersor agents, who represent distinct supply chain entities such as raw ma-terial suppliers, manufacturers, logistic firms, wholesalers, and/or retailers.The supply chain economy is, hence, a network of agents and their activities,

which can include, for example, the production, distribution, vendition, andconsumption of one or more products. Hence, a supply chain network econ-omy may consist of several supply chains which interact in their procurement,production, logistic, and retailing activities.

The notable features of the network model include:

1. The model is not limited to a fixed number of tiers with similar func-tions/operations associated with a given tier as in the work of Cachon andZipkin (1999), Cachon and Lariviere (1999a, 1999b), Nagurney, Dong, andZhang (2002), Nagurney et al. (2002a, b), Dong, Zhang, and Nagurney(2002). For an example of a supply chain network with a fixed number oftiers and similar functions associated with a given tier, see Figure 1. Themodel developed in this chapter, in contrast, allows for an entirely generalstructure of the supply chains. In addition, the model allows for the studyof inter chain competition as well as coordination and integration of intrachain activities with appropriately defined variables, network topology, andcosts associated with the links.

2. The model addresses inter supply chain competition at multiple marketsand does not assume homogeneity of the product throughout the chain but,rather, captures also the costs associated with the subcomponents as theproduct proceeds down the chain.

3. The network model includes operation links as well as interface links with

the former corresponding to such possible business operations as manufac-turing, storage, and transportation and with the latter acting as a bridgebetween business to business.

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n1 n j nn Retailers

n1 n i nm

n1 n h nr

n1 n k no

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A

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A c

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Manufacturers

Demand Markets

Suppliers

Figure 1: An example of a four-tiered supply chain network with suppliers,manufacturers, retailers, and demand markets and with similar functionsassociated with a given tier

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The cost on a specific interface link, in turn, can measure the effectivenessof the coordination and the integration of the network agents. Moreover,we impose no artificial conditions on the cost functions associated with thevarious links.

In Section 3, we derive the equilibrium conditions, which are based onthe identification of those chains in the network with minimal marginal chaincost and positive flow. The concept, which is based on the ideas of Wardrop(1952), and which has had wide application in traffic networks as well as otherapplication settings (see Nagurney (1999), Nagurney and Dong (2002), andthe references therein), is here generalized to apply to network chains, ratherthan paths, and to material flows, rather than identical types of network

flows (such as travelers). We establish the variational inequality formulationof the equilibrium conditions governing the supply chain network economy.It is noteworthy that the model enables us to bring potential supply chainstructures into our study. Moreover, the solution of the model identifiespotential supply chain structures with minimum marginal costs and, hence,predicts what supply chain may prevail in the future. Consequently, themodel can suggest effective supply chain formations.

In Section 4, we then turn to the examination of the qualitative propertiesof the equilibrium flow pattern and provide both existence and uniqueness re-sults. Section 5 summarizes the results of this paper and presents suggestionsfor future research.

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We consider two basic types of links in the network model: an operationlink and an interface link. An operation link represents a business functionperformed by a firm in a supply chain network, and can reflect a manufac-turing, transportation, storage, or, even, service operation. A firm may haveseveral operation links in the supply chain network economy with distinctlinks corresponding to the firms distinct business functions. An interfacelink, in turn, serves as a business to business bridge and lies, typically, be-tween operation links in the network model. If we denote the set of operationlinks by A and the set of interface links by B, then we have that L = AB.

We now distinguish between potential and active supply chains, with thelatter playing an especially important role in terms of competitiveness of the

supply chain network economy. A potential supply chain is represented by aconnected subgraph of the network characterized by a destination node corre-sponding to its pertinent market. It is recognized in the model that a possiblesupply chain structure may not necessarily be prevailing at a particular time.However, a potential chain can evolve to prevail in the supply chain economyif it possesses a competitive structure. A potential supply chain becomesactive in our model if it carries positive flow of material; hence, a potentialsupply chain is active if and only if its chain flow of material is positive.Later, we provide conditions that allow us to explicitly identify such chainsand chain flows.

Let Sj denote the set of all potential supply chains pertaining to market

j. Sdenotes the set of all potential supply chains in the network representingthe supply chain economy.

The chain flow of a supply chain is defined to be the volume of the finalor end product that the supply chain delivers to its pertinent market, thatis, it is the output of the supply chain. Let Xs denote the chain flow of chains and let X be the vector of all the chain flows of all the supply chains inthe supply chain economy. The volume of the end product of a supply chaindetermines the amount of work processed at each stage of the materialflow stream. For example, by tracking back the flow of material from theend product and exploding the series of bills of materials, one can determine

the amount of work (and information) processed or resources (such as laborhours, machine hours, storage space, etc.) utilized in each participator linkof a supply chain. To make such a measure rigorous, we define the processrate below.

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f

f

I

q

a21

a22

f Eb21 f

f

f

E

Ea11

a12

f

f

dd

d

b11

b12

f Efa13 Efb13d

dd

a14

I

a23

Demandmarket

Figure 2: An example of two supply chains competing at a demand market

of truck loads required for transportation, or even the number of squarefootage in the warehouse used for storage purposes.

Recall that Xs denotes the chain flow of chain s. Then, one has thefollowing relationship between an operation link flow and a chain flow:

xa = asXs, a As, s S. (2)

For an interface link b, on the other hand, the link flow xb indicatesthe amount of integration work, coordination effort, and/or informationprocessed on this link. Therefore, the link flow of an interface link is de-termined by the chain flow of its governing supply chain, that is,

xb = xb(Xs), b Bs, s S. (3a)

It is reasonable to assume that the chain-link flow relationship given in (3a) isa strictly monotone increasing function, although it can be nonlinear; that is,the greater the volume of the supply chain output, the greater the integrationwork required on all the interface links. Hence, the inverse function of (3)

exists, that is,Xs = X

1s (xb), b Bs, s S. (3b)

The link cost of an operation link is defined to be the total cost incurredby the firm in performing the corresponding task function, which is in concert

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with the definition of an operation link being a business function performedby a firm. Therefore, the link cost ca of link a is a function of its flow xa,and can be expressed as

ca = ca(xa), a A; (4)

in other words, the cost of carrying out a task function depends on theamount of work processed in the task. Depending upon the nature of theoperation corresponding to the link, the link cost function can be nonlinear,with consolidation and congestion being the two primary factors that deter-mine the cost structure. For example, if a batch or lot size is required for aproduction operation or if a truckload or shipload is applicable, a consolida-

tion effect can be expected to be present and, thus, the greater the volume,the cheaper the rate. On the other hand, if the link represents a bottleneckoperation, then one can expect a congestion effect, with the function beingincreasing in the volume of traffic.

The cost of an interface link, since such a link is an inter-link, reflectsthe effectiveness of coordination and integration of the two operation linksthat it bridges. The geographic distance, the experience of past cooperation,and/or the sophistication of the information system integrated in the two

joined operations, as well as the compatibility of the two, represent severalfactors that could account for the cost of an interface link. Since effectivecoordination of two successive operation links of a supply chain may dependupon other joints of this chain as well, we assume that the cost of an interfacelink may, in general, depend on the flows of all the interface links belongingto the same supply chain. However, in light of (3a), it can be determined bythe chain flow of its governing supply chain, and according to (3b), it alsouniquely depends on the link flow of the interface link, that is,

cb = cb(Xs) = cb(xb), b Bs, s S. (5)

The cost of a supply chain s is then the sum of the link costs of itsoperation and interface links, which can be expressed as

Cs = aAs

ca + bBs

cb. (6)

We assume that the demand for the product at a market place is elasticand can be characterized by its market value. Let qj and vj denote, respec-tively, the supply and the market value of the product at market j. Further,

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let q, and v be, respectively, the corresponding n-dimensional vectors of sup-plies and prices. By definition, the supply at a market is equal to the sum ofthe chain flows of all the supply chains that are driven by that market, thatis,

qj =sSj

Xs, j = 1, . . . , n . (7)

In general, we assume that the market value of the product at a marketmay depend on the vector of supplies of the products at all the markets, thatis,

vj = vj(q), j = 1, . . . , n , (8)

or, in vector form,

v = v(q) = (vj(q); j = 1, . . . , n). (9)

It is reasonable to assume, from an economic perspective, that, in (8) and(9), the market value is a decreasing function of the supply.

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Definition 4: Chain Marginal Cost

The chain marginal cost of a supply chain s is defined to be the unit chainprocessing cost of chain s, if the chain flow Xs is no less than one unit;otherwise, it is the chain cost for one unit of flow.

Cs =

Cs(Xs)/Xs, if Xs 1C(1), if Xs 1.

(12)

It is clear that the above defined link marginal costs and chain marginalcost are continuous when the link costs are continuous, and are differentiablewhen the link costs are differentiable. Furthermore, in view of (9.6), we canspell out the chain marginal cost in terms of link marginal costs in a uniformexpression regardless of the value of Xs:

Cs =aAs

asca(xa) +bBs

cb(Xs). (13)

An end market can be viewed as the destination of the product, which,in turn, pulls the flow of materials through all its pertinent supply chains.The supply chains, as alternative paths to the destination are racingto deliver the product. In line with the perspective that the firms goal interms of cooperation in the supply chain is to make the final product overall

at lesser cost than competing supply chain firms (see Covinato (1992)), thewinning chains in this race or competition are, in effect, the shortest paths,that is, those that can deliver the product at the least marginal cost.

There is an interesting analogy between this interpretation and that oc-curring in the context of traffic networks in which travelers seek to determinethe shortest paths from their origins to their destinations. In a congested ur-ban transportation network, travelers are assumed to select shortest paths(minimal cost paths) from their origins to their destinations with Wardrop(1952) being credited with the following (first) principle:

Wardrops First Principle

The travel costs of all paths actually used are equal and less than those whichwould be experienced by a single traveler on any unused path.

One should realize that the network structure of a supply chain can beany connected subgraph, with the simplest topology being that of a tree

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and, hence, it is more complex than a path in a transportation network froman origin to a destination. Also, unlike travelers (in vehicles), who arriveat their destinations in the same form as they leave their origins, the flowof material in a supply chain network economy evolves from raw materials,through fabricated parts, subassemblies, to finished products, compoundedat various stages of a supply chain process.

Nevertheless, the economic rationale that an inferior or less cost-effectivesupply chain will lose to its rivals is in the same spirit as Wardrops firstprinciple.

We now formally state the definition of an equilibrium in this framework.

Definition 5: Equilibrium of a Supply Chain Network Economy

In a supply chain network equilibrium, the active supply chains are thosewhose marginal cost is equal to the market value of the product at the perti-nent market, and the inactive supply chains are those whose marginal cost isgreater than or equal to the market value of the product at the pertinent mar-ket. Mathematically, a feasible supply chain network economy with flows X,with induced q through (7), constitutes a supply chain network equilibriumif and only if the following system of equalities and inequalities holds true:

Cs(X)

= vj(q

), if Xs > 0 vj(q

), if Xs = 0,s Sj, (14)

for all end markets j; j = 1, . . . , n.

With this definition, we now have the answers to the aforementionedquestions. The winning (active) chains are those supply chains with leastmarginal cost for their pertinent markets, and they are the only ones thatcarry positive flow of materials.

Before deriving the variational inequality formulation of the governingequilibrium conditions, we introduce link capacities. Recall that since anoperation link may represent a manufacturing process, transport, and/orstorage, it may face such constraints as limited machine hours, a fixed numberof trucks available, and/or limited storage space in a warehouse. Denote the

link capacity of an operation link by ha. Then, we have that

asXs = xa ha, a As, s S. (15)

The feasible set underlying the supply chain economy in terms of the

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chain variables accounting operational capacities is expressed as

X = {X 0 : (9.15) holds for all a As, s S} . (16)

This set is equivalent to the feasible set in chain variables and supplypatterns, defined as

Xq = {X 0, q 0 : (7), (15) hold for alla As, s S; j = 1, . . . , n} .(17)

The following theorem gives a mathematical formulation for the supplychain network equilibrium in chain variables.

Theorem 1: Variational Inequality FormulationX with induced q through (7) is a supply chain network equilibrium if andonly if(X, q) solves the following variational inequality problem: determine(X, q) Xq such that

nj=1

sSj

Cs(X)(Xs X

s ) vj(q)(qj q

j )

0, (X, q) Xq . (18)

Proof: The proof is standard (see, e.g., Nagurney (1999)). 2

3.1 An Example

In this subsection, we present a numerical example. Consider a sup-ply chain network example as depicted in Figure 3. There are two demandmarkets, Market 1 and Market 2. There are four supply chains, denoted,respectively, by S1, S2, S3, and S4 and also referred to, without loss of gen-erality, as Supply Chains 1, 2, 3, and 4. Supply chain S1 is comprised ofoperation links a11, a12, a13 and interface link b11. S2 is comprised of oper-ation links a21 and a22 and interface link b21. S3, in turn, is comprised ofoperation links a31, a32, and a33 and interface links b31 and b32. Supply chainS4 consists of operation links a41, a42, a43, and a44 and interface links b41, b42,and b43. Supply Chain 1 and Supply Chain 3 are competing for Market 1,while Supply Chain 2 and Supply Chain 4 are competing for Market 2.

The process rates for the operation links are given by:

11 = 2, 12 = 3, 13 = 1,

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e eEU

a42

e

B

rrrrrj

b42

b41 e eEU

a41eE

a43

eE

b43

E

a44

Market 2

e

ee

e

q

I

b31

b32ez

a33

a31 a32

Ea12

d

dddd

a11

e Eb12

eza13

za21

ezb21

eza22

Market 1

Figure 3: An example of a Supply Chain Network Economy

21 = 4, 22 = 1,

31 = 2, 32 = 2, 33 = 1,

41 = 2, 42 = 4, 43 = 2, 44 = 1.

The link marginal cost functions of the operation and the interface links,for each supply chain, are as given below.

ca11 = 0.5xa11 + 1, ca12 = 0.1xa12,

cb11 = 2.1xb11 + 1, ca13 = 5xa13 + 3,

ca21 = 0.5xa21 + 0.5, cb21 = X2, ca22 = xa22 + 2,

ca31 = xa31 + 4, ca32 = xa32 + 4, cb31 = X3 + 4,

cb32 = X3 + 4, ca33 = xa33 + 4,

ca41 = ca42 = ca43 = 0.5,

ca44 = 2xa44 + 1, cb41 = cb42 = cb43 = X4.

The chain marginal costs can be computed according to (13) by using thelink marginal costs. They are as follows:

CS1 = 10X1 + 6, CS2 = 10X2 + 4, CS3 = 11X3 + 28, CS4 = 5X4 + 5.

Suppose now that the two demand markets are interrelated and that theirmarket value functions are:

v1(q) = 40 5q1 q2, v2(q) = 60 q1 10q2.

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The supply chain network equilibrium in this example is given by

X = (X1 , X

2 , X

3 , X

4 ) = (2, 1.4, 0, 2.6),

which generates the equilibrium supply pattern

q = (q1, q

2) = (2, 4).

The market value at Market 1 at equilibrium is 26, whereas the market valueat Market 2 at equilibrium is 18.

Examining the two competing supply chains for Market 1 at the equilib-rium, one see that the marginal chain cost of Supply Chain 1 is 26 and the

marginal chain cost of Supply Chain 3 is 28. This verifies the equilibriumcondition that the market value is equal to the marginal chain cost for theactive (winning) supply chain, Supply Chain 1, and that the inactive chain(the loser), Supply Chain 3, has a marginal cost that is higher than the mar-ket value. The equilibrium condition also holds for Market 2. In this case,the two competing supply chains, Supply Chain 2 and Supply Chain 4, areboth active, and have the same marginal chain cost, 18, which is equal tothe market value at Market 2 in equilibrium.

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4. Qualitative Properties

This section provides the fundamental qualitative properties of the math-ematical formulation of the equilibrium of the supply chain network economypresented in the previous section. First, we establish the existence of an equi-librium and then turn to its uniqueness.

Theorem 2: Existence

Suppose that the link cost functions for all the operation and interface linksare continuous. Then, there exists an equilibrium of the supply chain networkeconomy.

Proof: We claim that the feasible set Xq is compact. It is easy to see that

Xq is closed. To show that it is bounded, one has, in view of (15), for everysupply chain s

Xs 1as ha, a As, (19)

which yieldsXs min

aAs{1as ha}, s S. (20)

(20) suggests that in (7),

qj sSj

minaAs

{1as ha}, j = 1, . . . , n . (21)

Expressions (20) and (21) imply that all the variables are, indeed, boundedfrom above, and, hence, Xq is bounded.

As noted before, the continuity of the link cost functions guarantees thecontinuity of the link marginal cost functions, which further implies the con-tinuity of the chain marginal cost functions through (13). Therefore, thevariational inequality (18) is a continuous mapping over a compact set. It,thus, according to the standard theory of variational inequalities (see Kinder-lehrer and Stampacchia (1980)) has at least one solution. 2

The uniqueness of the supply chain network equilibrium pattern can, ingeneral, be ensured under an assumption of strict monotonicity (cf. Nagurney(1999) and the references therein) of the marginal link cost functions and the

market value functions as given by the following theorem.

Theorem 3: Uniqueness

Suppose that the marginal link cost functions are strictly monotone increas-ing, and the market value functions are strictly monotone decreasing, with

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respect to their arguments. Then, there exists a unique equilibrium of thesupply chain network economy.

Proof: The monotonicity assumption on link marginal cost functions impliesthat the chain marginal cost function is monotone increasing, in view of(13). Therefore, under the condition of the theorem, the vector functionof the variational inequality problem (18) is monotone. The conclusion ofthe theorem follows from Theorem 2 and the standard theory of variationalinequality problem (see, e.g., Nagurney (1999)). 2

Let Jqv denote the Jacobian matrix of the market value function withrespect to the supply vector q. It is well-known (see, e.g., Nagurney (1999))

that ifJqv is symmetric, then there is a real-valued function v whose gradientis v, namely,v = qv. (22)

A special case of v being a gradient of a real-valued function is that themarket value function is separable; namely, in this case, the market value ofthe product at market j depends only on the supply at market j. In thiscase, the Jacobian matrix ofv is a diagonal matrix with the diagonal entriesbeing the derivative of the market value with respect to their own supply.

Under certain conditions, the supply chain network equilibrium can alsobe formulated as the following optimization problem:

min(X,q)Xq

nj=1

sSj

Xs

0Cs(u)du v(q) (23)

Theorem 4: Sufficient Condition

Suppose that the Jacobian matrix of the market value function is symmetric.Then, a sufficient condition for (X, q) to be an equilibrium of a supplychain network economy is that it solves the optimization problem (23).

Proof: According to Theorem 1, it suffices to show that any solution tooptimization problem (23) is a solution to (18). However, the variational

inequality problem (18) is simply a restatement of the first-order necessarycondition of the optimization problem (23) by noticing that the vector func-tion entering into (18) is the gradient of the objective function in (23). 2

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Theorem 5: Necessary and Sufficient Condition

Suppose that the marginal link cost functions are monotone increasing, andthe Jacobian matrix of the market value function is symmetric and negativesemi-definite. Then, (X, q) is an equilibrium of the supply chain networkeconomy if and only if it solves the optimization problem (23).

Proof: The vector function entering (18) is the gradient of the objectivefunction in (23). Hence, the assumption of monotone marginal link costs, to-gether with negative semi-definiteness of the Jacobian matrix of the marketvalue function, implies that the objective function of (23) is convex. Ac-cording to Bazaraa, Sherali, and Shetty (1993), the variational ienqualityproblem (23) is the necessary and sufficient condition of the solution to the

convex programming (23). 2

5. Summary and Conclusions

In this paper, we proposed a new framework, that is network-based, toformalize the modeling and analysis of supply chains. The framework allowsfor the simultaneous study of both intra supply chain cooperation as well asinter supply chain competition. Moreover, it has the notable feature thatunlike many of the supply chain models in the literature, the underlyingnetwork topological structure is general. In addition, the model allows forthe transformation of material flows as they proceed through the network.

We developed the supply chain network economic model, derived the gov-erning equilibrium conditions, and then provided the variational inequalityformulation. We established existence of an equilibrium pattern, and alsogave conditions under which uniqueness is guaranteed.

Acknowledgments

The research of the second and third authors was supported, in part, byNSF Grant No.: IIS-0002647. The research of the third author was also sup-ported by an AT&T Industrial Ecology Fellowship. The support is gratefullyacknowledged. The authors are grateful to the anonymous reviewers for theircomments and suggestions on an earlier version of this manuscript.

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A Supply Chain Network Economy Modeling and Qualitative Analysis

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